Milica Andelić, Carlos M. da Fonseca, António Pereira: The μ-permanent, a new graph labeling, and a known integer sequence, 255-262

Abstract:

Let $ A=(a_{ij})$ be an $ n$-by-$ n$ matrix. For any real number $ \mu$, we define the polynomial

$\displaystyle P_\mu(A)=\sum_{\sigma\in S_n}
a_{1\sigma(1)}\cdots a_{n\sigma(n)}\,\mu^{\ell(\sigma)}\; ,$

as the $ \mu$-permanent of $ A$, where $ \ell(\sigma)$ is the number of inversions of the permutation $ \sigma$ in the symmetric group $ S_n$. In this note, motivated by this notion, we discuss a new graph labeling for trees whose matrices satisfy certain $ \mu$-permanental identities. We relate the number of labelings of a path with a known integer sequence. Several examples are provided.

Key Words: $ \mu$-permanent, $ q$-permanent, determinant, permanent, graph, tree, path, graph labeling, integer sequence, Mathematica

2010 Mathematics Subject Classification: Primary 15A15. Secondary 05C50, 05C78, 05C30, 68R10, 11B83